AP®︎ Statistics: Topic 4.2 Flashcards
Master key terms and definitions for Topic 4.2 of AP Statistics – Constructing a Confidence Interval for a Population Mean or Population Mean Difference to help you prep for quizzes and the AP exam.
Student’s t-Distributions
Family of symmetric, bell-shaped, standardized distributions centered at 0 with lower peaks and heavier tails than the standard normal; identified by degrees of freedom
Family of symmetric, bell-shaped, standardized distributions centered at 0 with lower peaks and heavier tails than the standard normal; identified by degrees of freedom
Parameter that identifies a t-distribution; for a one-sample t-procedure, df = n − 1
Use t when making inference about a population mean and σ is unknown, so s is used instead, adding uncertainty beyond z-procedures
Positive t value with central C% of the t-distribution between −t* and t*; depends on confidence level and df
Confidence interval procedure for estimating μ from one sample of quantitative data when σ is unknown
Confidence interval procedure for estimating μ_d, the population mean of paired differences, by applying a one-sample t-interval to one sample of within-pair differences
Design with two related measurements for each individual or matched pair; analyze by computing within-pair differences and treating those differences as one sample
The population mean of the quantitative response variable, stated in context
The population mean of the paired differences, stated in context with the subtraction order defined
For paired differences, define d = A − B; reversing the order changes the sign of every difference, d̄, and both interval endpoints, but not the standard deviation
Data must come from a random sample or a randomized experiment
When sampling without replacement from a finite population, require n ≤ 0.10N, or equivalently N ≥ 10n
For a one-sample t-interval, the population is approximately normal, or n ≥ 30, or if n < 30 the sample distribution is free from strong skewness and outliers
Check the distribution of the differences: either at least 30 differences, an approximately normal population of differences, or with fewer than 30 differences no strong skewness or outliers
x̄, the sample mean
d̄ or x̄_d, the sample mean of the paired differences
x̄ ± t*(s/√n), with df = n − 1
d̄ ± t*(s_d/√n), with df = n − 1 and n equal to the number of pairs or differences
SE = s/√n; estimated sample-to-sample variability of x̄
ME = t* × SE = t*(s/√n)
s measures variability among individual observations; s/√n estimates variability among sample means
t* is a standardized cutoff from the t-distribution; margin of error is t* times SE and is in the response variable’s units
Do not use a two-sample t-interval when observations are paired; compute within-pair differences and use a one-sample t-interval on the differences
